Unfortunately, the domain dropdown is gone in all Windows releases after Windows XP and there is nothing that can be done about it. The best solution I have found so far is to type .\username to login locally... by using the full stop at least then you don't need to remember the computer name.
There was also a NoDomain registry value IIRC. You could try that.
1. In the Registry Editor, navigate to HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Winlogon
2. Create a new DWORD of NoDomainUI
3. Set the value of this DWORD to 1 (Enabled)
4. Reboot the machine
If the NoDomainUI value is enabled,
Windows will always assume you are trying to logon to the local computer. To logon to the domain, it will force users to use the full UPN (user principal name): or domain\username. Does that help?